Algebraic integer

Algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
The ring of integers of a number field , denoted by , is the intersection of and : it can also be characterised as the maximal order of the field . Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring is finitely generated as an abelian group, which is to say, as a -module.
Definitions
The following are equivalent definitions of an algebraic integer. Let be a number field (i.e., a finite extension of , the field of rational numbers), in other words, for some algebraic number by the primitive element theorem.
is an algebraic integer if there exists a monic polynomial such that .
is an algebraic integer if the minimal monic polynomial of over is in .
is an algebraic integer if is a finitely generated -module.
is an algebraic integer if there exists a non-zero finitely generated -submodule such that .
Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension .
Examples
The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of and is exactly . The rational number is not an algebraic integer unless divides . Note that the leading coefficient of the polynomial is the integer . As another special case, the square root of a nonnegative integer is an algebraic integer, but is irrational unless is a perfect square.
If is a square-free integer then the extension is a quadratic field of rational numbers. The ring of algebraic integers contains since this is a root of the monic polynomial . Moreover, if , then the element is also an algebraic integer. It satisfies the polynomial where the constant term is an integer. The full ring of integers is generated by or respectively. See Quadratic integer for more.
The ring of integers of the field has the following integral basis, writing for two square-free coprime integers and :
If is a primitive th root of unity, then the ring of integers of the cyclotomic field is precisely .
If is an algebraic integer then is another algebraic integer. A polynomial for is obtained by substituting in the polynomial for .
Non-example
If is a primitive polynomial which has integer coefficients but is not monic, and is irreducible over , then none of the roots of are algebraic integers (but are algebraic numbers). Here primitive is used in the sense that the highest common factor of the coefficients of is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.
Facts
The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher degree than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if , and , then eliminating and from and the polynomials satisfied by and using the resultant gives , which is irreducible, and is the monic equation satisfied by the product. (To see that the is a root of the -resultant of and , one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)
Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not. This is the Abel–Ruffini theorem.
Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring which is integrally closed in any of its extensions.
The ring of algebraic integers is a Bézout domain, as a consequence of the principal ideal theorem.
If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the reciprocal of that algebraic integer is also an algebraic integer, and is a unit, an element of the group of units of the ring of algebraic integers.
Every algebraic number can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer. Specifically, if is an algebraic number that is a root of the polynomial with integer coefficients and leading term for then is the promised ratio. In particular, is an algebraic integer because it is a root of , which is a monic polynomial in with integer coefficients.
See also
Integral element
Gaussian integer
Eisenstein integer
Root of unity
Dirichlet's unit theorem
Fundamental units
References
Algebraic numbers
Integers